Propagation and heterogeneous steady states in a delayed nonlocal R-D equation without spatial translation-invariance

Abstract

We consider a delayed reaction-diffusion equation that models the population dynamics of a single species with the mature population living in the 1-D whole space R while the immature population only living in the half space R+, with homogeneous Dirichlet condition for the immatures at the boundary point. One of the important features of this model system is that it does have the translational-invariance. By linking the non-translational-invariant solution map for this equation to travelling wave maps for another related 1-D spatial homogeneous delay reaction-diffusion equation, we obtain some traveling-like a priori estimates for nontrivial solutions. We then establish the existence, uniqueness, and attractivity of heterogeneous steady states. As a result, we are able to describe the traveling-like asymptotic behaviours of nontrivial solutions in space-time region. These enable us to develop a new method for exploring the spreading speeds and asymptotic propagation phenomena for a class of non-translation-invariant delay reaction-diffusion equations on R. As a corollary, we also recover some results on the asymptotic spreading speeds and traveling waves for monostable and spatial homogeneous delay reaction-diffusion equations in R.